0
votes
1answer
18 views

Writing down the transition matrix of a discrete Markov chain

Please consider the following scenario: One person is walking along a discrete circle induced by $\mathbb{Z}/n\mathbb{Z}$ In each round we roll a dice with $w\in\left\{2,\ldots n\right\}$ sides If ...
0
votes
0answers
69 views

proving null recurrence of random walk (Markov chain)

How would I prove that the zero state of a random walk with a positive probability of staying in the same state is null recurrent. (sorry if this isn't a random walk and just a Markov chain.) eg. ...
3
votes
1answer
64 views

Random walk where increment depend on current position

Consider the following stochastic process, $$b(i+1) = b(i) + \xi_i (b_i),$$ where $\xi_i(b_i) \in \{-1, k \}$ are the independent increments having the following distribution: $$\begin{align} P (\xi ...
4
votes
1answer
77 views

random walk in a certain environment

Consider the following random walk in one dimension, starting from $r(0)=0$. $$ r(i+1) = r(i) + \xi, $$ where $\xi(i, r(i))$ is an increment with distribution $P(\xi=1) = \frac{c^{r(i)}}{i-r(i)+1}$ ...
1
vote
0answers
25 views

Prove equilibrium theorem without irreducibility and aperiodicity

I have to solve the following question: Consider a random walk Markov chain on $S = \{1, 2, \ldots, 100\}$. If the chain is between 2 and 99, it selects one of the adjacent states with equal ...
0
votes
0answers
26 views

Why must a stochastic process be at least second order in terms of differential equations?

A first order differential equation in $q(t)$ has a unique path through each possible value of $q(0)$. This is opposed to a stochastic process (e.g. random walk), where any place might be "hopped ...
1
vote
1answer
63 views

Absorbing state for a collection of random walks

Further to this question; having learned some stuff since I posed it. Consider a collection of random walks $X_i$ which take finite integer values. These evolve as time-inhomogeneous Markov Chains. ...
4
votes
0answers
128 views

Conditional probability and integrating out part of a random walk

Suppose that I have a random walk process defined by $\alpha_{t+1}$ ~ N$(\alpha_t, \omega^2)$. Given $\alpha_t$ and $\alpha_{t+2}$, I understand why the conditional formula for ...
0
votes
1answer
128 views

geometric sum - weighted random walk

I am trying to model the following sum: $\sum_{i=0}^{n}{W_i \alpha^{i}}$ where $\alpha \in[0, 1) $ and $W_n$ takes values 0 or 1 and may be modeled as a markow chain or for simplicity as a binary ...
4
votes
1answer
337 views

Covariance of Brownian-motion-like processes

We know that $\operatorname{Cov}(B_s,B_t)=\min(s,t)$ if $B_t$ is Brownian motion. What is $\operatorname{Cov}(B_{f(s)},B_{f(t)})$ for some injective $f$? How can I write $B_{f(t)}$ in an Ito ...
3
votes
3answers
106 views

Introduction to Markov Random Fields

I'm looking for a gentle introduction to this topic. The material I've found so far is substantially related to physics, and requires some background in such field. Is there anything more general and ...
12
votes
3answers
274 views

Select a new value from last $N$ values; how long until the last $N$ are all the same?

Say first we have N distinct numbers in a line, like 1,2,3,...,N, in each round, we choose a ...